## Zero Number (0)

## Zero number definition

Zero is a number used in mathematics to describe no quantity or null quantity.

When there are 2 apples on the table and we take the 2 apples, we can say that there are zero apples on the table.

The zero number is not positive number and not negative number.

The zero is also a placeholder digit in other numbers (e.g: 40,103, 170).

## Is zero a number?

Zero is a number. It is not positive nor negative number.

## Zero digit

The zero digit is used as a placeholder when writing numbers.

For example:

204 = 2×100+0×10+4×1

## Zero number history

## Who invented the zero number?

The modern 0 symbol was invented in India in the 6-th century, used later by the Persians and Arabs and later in Europe.

## Symbol of zero

The zero number is denoted with the 0 symbol.

The Arabic numeral system uses the ٠ symbol.

## Zero number properties

x represents any number.

Operation Rule Example ## Addition

x+ 0 =x3 + 0 = 3

## Subtraction

x- 0 =x3 - 0 = 3

## Multiplication

x× 0 = 05 × 0 = 0

## Division

0 º

x= 0 , whenx≠ 00 º 5 = 0

xº 0 is undefined5 º 0 is undefined

## Exponentiation

0

= 0^{ x}0

^{5}= 0

x^{0}= 15

^{0}= 1## Root

√0 = 0

## Logarithm

log

_{b}(0) is undefined## Factorial

0! = 1

## Sine

sin 0º = 0

## Cosine

cos 0º = 1

## Tangent

tan 0º = 0

## Derivative

0' = 0

## Integral

∫ 0 d

x= 0 +C## Zero addition

Addition of a number plus zero is equal to the number:

x+ 0 =xFor example:

5 + 0 = 5

## Zero subtraction

Subtraction of a number minus zero is equal to the number:

x- 0 =xFor example:

5 - 0 = 5

## Multiplication by zero

Multiplication of a number times zero is equal to zero:

x× 0 = 0For example:

5 × 0 = 0

## Number divided by zero

Division of a number by zero is not defined:

xº 0 is undefinedFor example:

5 º 0 is undefined

## Zero divided by a number

Division of a zero by a number is zero:

0 º

x= 0For example:

0 º 5 = 0

## Number to the zero power

The power of a number raised by zero is one:

x^{0}= 1For example:

5

^{0}= 1## Logarithm of zero

The base b logarithm of zero is undefined:

log

_{b}(0) is undefinedThere is no number we can raise the base b with to get zero.

Only the limit of the base b logarithm of x, when x converges zero is minus infinity:

## Sets that contain zero

Zero is an element of the natural numbers, integer numbers, real numbers and complex numbers sets:

Set Set membership notation Natural numbers (non negative) 0 ∈ ℕ _{0}Integer numbers 0 ∈ ℤ Real numbers 0 ∈ ℝ Complex numbers 0 ∈ ℂ Rational numbers 0 ∈ ℚ ## Is zero even or odd number?

The set of even numbers is:

{... ,-10, -8, -6, -4, -2,

0, 2, 4, 6, 8, 10, ...}The set of odd numbers is:

{... ,-9, -7, -5, -3, -1,

1, 3, 5, 7, 9, ...}Zero is an integer multiple of 2:

0 × 2 = 0

Zero is a member of the even numbers set:

0 ∈ {2

k,k∈ℤ}So zero is an even number and not an odd number.

## Is zero a natural number?

There are two definitions for the natural numbers set.

The set of non negative integers:

ℕ

_{0}= {0,1,2,3,4,5,6,7,8,...}The set of positive integers:

ℕ

_{1}= {1,2,3,4,5,6,7,8,...}Zero is a member of the set of non negative integers:

0 ∈ ℕ

_{0}Zero is not a member of the set of positive integers:

0 ∉ ℕ

_{1}## Is zero a whole number?

There are three definitions for the whole numbers:

The set of integer numbers:

ℤ = {0,1,2,3,4,5,6,7,8,...}

The set of non negative integers:

ℕ

_{0}= {0,1,2,3,4,5,6,7,8,...}The set of positive integers:

ℕ

_{1}= {1,2,3,4,5,6,7,8,...}Zero is a member of the set of integer numbers and the set of non negative integers:

0 ∈ ℤ

0 ∈ ℕ

_{0}Zero is not a member of the set of positive integers:

0 ∉ ℕ

_{1}## Is zero an integer number?

The set of integer numbers:

ℤ = {0,1,2,3,4,5,6,7,8,...}

Zero is a member of the set of integer numbers:

0 ∈ ℤ

So zero is an integer number.

## Is zero a rational number?

A rational number is a number that can be expressed as the quotient of two integer numbers:

ℚ = {

n/m;n,m∈ℤ}Zero can be written as a quotient of two integer numbers.

For example:

0 = 0/3

So zero is a rational number.

## Is zero a positive number?

A positive number is defined as a number that is greater than zero:

x> 0For example:

5 > 0

Since zero is not greater than zero, it is not a positive number.

## Is zero a prime number?

The number 0 is not a prime number.

Zero is not a positive number and has infinite number of divisors.

The lowest prime number is 2.

## See also